Remark 1 In more theoretical studies on controllability issues for such coupled parabolic systems, see survey [1]}, the control term is usually of the form \(B 1_{\omega } \bar{u}(t,x)\) , with control \(\bar{u}(\cdot , \cdot )\)  time and space-dependent and \(\omega \)  a given open subset of \([0,L]\) . We could have alternatively posed the present problem in this setting, however, in engineering problems, shape functions \(b_j(x)\)  are already given (see for instance [2]}) and we seek for stabilizing actuations depending exclusively on time as in the present analysis. In the first scenario, we would choose a proportional-type control (see [3]} (Chapter 2), see also [4]} (Chapter 9)) of the form
\(\bar{u}(t,x)=\sum _{j=1}^N \bar{K}_j b_j(x) \int _0^L z(t,x) \varphi _j(x) dx,\) 
where \(\bar{K}_j \in \mathbb {R}^{1\times m}\)  and functions \(b_j(\cdot )\)  are chosen to be written as
\(b_j(x)=\sum _{k=1}^N b_{jk} \varphi _k(x), \quad j=1,\ldots , N, \) 
with coefficients \(b_{jk} \in \mathbb {R}\)  satisfying
\( \sum _{k=1}^N b_{jk} \int _0^L 1_{\omega } \varphi _k(x) \varphi _n(x) dx=\delta _{jn},\)  for all \( j, n =1,\ldots ,N.\) 
The previous equation satisfied by \(b_{jk}\) , thanks to the linear independence of eigenfunctions \(\varphi _n\) , leads to a unique solution written as
\(\mathcal {B}_{N\times N}= \Phi _{N\times N}^{-1},\) 
where \(\mathcal {B}_{N\times N}\)  is again defined as in (REF ) and \( \Phi _{N\times N}\)  is an \(N\times N\)  matrix with elements \((i,j)\)  given by \(\int _0^L 1_{\omega }\varphi _i(x)\varphi _j(x) dx\) . Then, the finite-dimensional part satisfies the same equations as in (REF ). More details about this alternative realization will be given in our future works.

